Based on a lecture by Rabbi David Fohrman(Cycle of Moadim pt.5):
I think the soul is my conscious mind- my I- my [sense of] self- my ego.
I don't think it's any mysterious thing.
If you ask me who am I- or where am I- I would say I feel like a cloud hovering over my forehead.
Interestingly, this is where G-d was in the Tabernacle/Temple. He was above the Ark cover between the Cherubim.
This Tabernacle/Temple could be schematized as a face, with the brain being the Ark(inside was the essence of the "Torah") and the mind being the G-d who hovers over the Ark; the eyes would be the Menora and the face-bread table(that which is illuminated); the nose would be the Golden insence altar; the mouth would be the altar of burnt offerings(God's "food").
G-d is saying to us, "You want to know where I am? I am in the same place you are: a mind hovering above the brain, on a face."
By hovering above the Ark, G-d is saying that- just like the your mind- He doesn't inhabit time and space either.
So the Tora teaches us how to think of/locate our souls: the soul is another name for the mind that hovers above the brain.
There are brain studies which corroborate this- I mean, the transcendence of the mind in brain studies.